Different species concepts exist because there is some debate about exactly how a species should be defined in conjunction to another; for example, there are birds that look nearly identical, but are considered to be different species, because of geographical location, or different mating habits, but members of the same species can look completely different from one another, like black versus grizzly bears, or male versus female peacocks. The concepts serve to define one school of thought regarding species in the latter category.
None of the concepts is any more correct than the others, because they each serve a particular purpose. Researchers may pick one concept to use to focus their work on a particular subject. The concepts approach the matter of connections between species in a number of ways, from shared ancestry, to the ability to recognize another as a potential mating partner (even if they are geographically separated, and unable to mate in practice).
A speciation is an event in which species come together, particularly in a way that would not be possible without intervention, or of thinning populations that drive species together when they normally would not have an opportunity to mate, that produces hybrid offspring. The hybrid offspring are counted as new species, which can potentially mate with either of the parent species. From there, the offspring species can create more hybrid lineages, and can create a slowly evolving line of offspring lineages. The forms of speciation are defined by geographically isolated populations, fringe populations, mixed populations like the canis latrans, and offspring mixing with the parent populations; they all create new offspring lineages when they mate.